Can I reduce the "number of weights" in CNN to 1/3 by restricting the input as greyscale image?
After inner product, add them all to make one feature map. Am I right?


yes, you are right.


  Then, can I reduce the number of weights in the filters? Because in this case, using three different n×n filters and adding them is same with using one n×n filter that is the summation of three filters.


If you have transformed the image into greyscale then you no longer need 3 filters. You should retrain your model on greyscale images. In a greyscale image the value of each pixel is a single sample representing only an amount of light (the light intensity).

The network will run faster if that is the only architectural change you make, but keep in mind that by converting the image to greyscale you will lose information and probably some of the predictive power of your network.